Hysteresis of the drag force of an intruder moving into a granular medium

We numerically investigate the force-displacement relation of a moving intruder initially at rest into a granular medium. Our model granular medium is composed of one layer of coplanar polydisperse spheres subjected to a gravity field. The interactions between the grains are modelled by Hertzian contacts to which a viscous damping is applied. Moving it horizontally and with alternating positive and negative velocity, we recover a hysteresis of the force-displacement curve. Considering that the flow is plastic as the yield strength has been reached, we describe the transient part of the flow around the intruder. We show that the drag stress increases as its distance to an ultimate drag stress sigma u with a typical deformation c: the drag stress-strains curve appears to exponentially decay as it saturates to this ultimate drag stress. This protocol of deformation highlights that the deformation of the grains is negligible compared to the deformation of the packing, i.e. related to the irreversible displacements of grains allowing the intruder to pass through. Simultaneously, the lift force is constant on average during the displacement of the intruder. We then give the different scaling laws of the yield strength, this ultimate drag stress, the characteristic deformation of the packing and the lift stress. Finally, we recover the complete hysteresis cycle of the drag force around the intruder.